The Dirac conjecture [1]} says that all first class constraints (whether they are primary, secondary, etc., or a combination of primary, secondary, etc.) generate gauge transformations. This conjecture does not hold as a general theorem—there are known examples in which the transformation generated by a secondary first class constraint does not coincide with any invariance of the original Lagrangian system.Counterexamples to the Dirac conjecture are discussed in Refs. [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]} and elsewhere. Proofs of the conjecture have been constructed by adopting various simplifying assumptions [10]}, [11]}, [3]}.
The status of the conjecture is a subtle issue; see for example Refs. [13]}, [14]}, [15]}. Nevertheless, the Dirac conjecture is usually taken as an assumption. It appears that in practice, for systems of physical interest, all first class constraints generate gauge transformations.
